Active Control of Flow Over a Backward-Facing Step

Transcription

1 Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 5 Seville, Spain, December -5, 5 hb. Active Control of Flow Over a Backward-Facing Step Abbas Emami-Naeini, Sarah A. McCabe, Dick de Roover, Jon L. Ebert, Robert L. Kosut Abstract We describe a control-oriented model reduction process from partial differential equation (PDE) descriptions of aerodynamic flow systems, and the design of low-order feedback controllers using these procedures. An effective approach for flow model reduction for active control using balanced truncation approaches was developed. he techniques were applied to a flow control problem representative of aerospace problems: active control of flow over a modified D backward-facing step, which can be handled by most commercial finite element packages. Although the geometry is rather simple, the richness of the flow field is well documented. Feedback controllers were developed for disturbance rejection and tracking of the reattachment point for both high and low Reynolds numbers. I. INRODUCION During the past two decades, there has been substantial effort to develop methodologies and tools to actively control fluid flow phenomena []-[4]. Possible applications include separation control, drag reduction, lift enhancement, and virtual aerodynamic shaping. he effort has consisted of analytical approaches, numerical simulations, and experimental work. he focus on active control of fluid dynamics has resulted in application of control-theoretic approaches to nonlinear partial differential equations (PDEs), and development of low-order physical models for control design. he development of physics-based lowdimensional models is a promising avenue toward active flow control [6]-[], [5]. In this paper, we describe feedback control design of active flow control based on reduced order models from balanced truncation and system identification. he method was applied to a problem that is representative of aerospace applications: flow over a modified D backward-facing step []-[4]. Nonlinear models were developed for both high and low Reynolds numbers that are three orders of magnitude apart. Feedback control designs were carried out for both disturbance rejection and tracking of the commanded reattachment point location for both high and low Reynolds number cases. he feasibility of the methods was demonstrated by linear and nonlinear simulations. For the characteristics of the flow we considered a plant model of order only four is sufficient to capture the control relevant dynamics. Manuscript received March 4, 5. his work was supported in part by AFOSR under Contract F496--C-. he program monitor was Lt. Col. Sharon Heise, USAF, Ph.D. he authors are with SC Solutions, Inc., Systems & Control Division, 6 Oakmead Pkwy, Sunnyvale, CA 9485 USA ( emami@scsolutions.com) /5/$. 5 IEEE 766 II. CASE SUDY:FLOW OVER A BACKWARD-FACING SEP Separation and reattachment flow occurs in many aerospace related systems such as diffusers, combustors, etc. Among the flow geometries used to study separated flows, the backward-facing step is common []-[4]. he flow over a backward-facing step has a simple geometry and also has flow characteristics similar to fully stalled diffuser flow []-[]. Even though the geometry is simple, the flow involves complicated streamlines as shown in Figure []. he separation is fixed by a sharp corner as shown. herefore, the separation-reattachment process can be examined by itself. he boundary layer separates at the edge of the step, and a free-shear layer develops inside the original shear layer. Part of the flow is reversed toward the step and re-entrained into the shear layer in the reattachment zone, which results in the formation of a recirculating zone behind the step. A new boundary layer grows on the surface downstream of the reattachment point. his layer then interacts with the approaching shear layer that flows over it from upstream. he separation-reattachment process is thus characterized by a complex interaction between the separated shear layer and the adjacent flow []-[4]. he location of the reattachment point is a good indicator of the dynamic flow behavior, and control of the location of this point can be used to control overall flow characteristics. In this paper, we discuss active control of the location of this point by suction and blowing. U Corner Recirculation Of Opposite Sign Free Shear Layer Recirculation Wall Boundary Layer Inviscid Core Dividing Streamline Reattachment Zone New Wall B. L. Figure. Schematic of streamlines in separationreattachment process []. A specific flow example (illustrated in Figure ) was used to demonstrate model reduction for separation control. his is the classical example of turbulent flow over a backwardfacing step with a modification representative of boundary flow control in an aerospace system. In this example, a flow with mean velocity v enters the channel from the left and passes over a backward-facing step. Optional suction or blowing control, u, on the step face allows control of

2 Entry Flow Boundary Layer Boundary Layer Suction or blowing to control reattachment Recirculation Reattachment Point Free Shear Layer Figure. urbulent flow over a backward-facing step with controls. important aspects of the flow, e.g., the location of the reattachment point. he addition of blowing or suction on the step face allows control of the position where the flow reattaches. If fluid is injected on the step (blowing u > ) the reattachment will move downstream, while suction (u < ) will draw it upstream. he relation between flow/velocity sensor locations and flow control power (under feedback) can be studied with this model. In addition, by varying the entry flow velocity, v, the model can be used to study control of transient flow. he specific benchmark example (with dimensions and parameters) is shown in Figure. linear model generation for the backward-facing step case study. III. NONLINEAR MODEL FOR FLOW OVER A BACKWARD- FACING SEP Nonlinear models were developed using the commercial software ADINA for both high and low Reynolds numbers that are three orders of magnitude apart. ADINA is a general purpose finite element commercial software package for linear and nonlinear analysis of structural, heat transfer, field and fluid flow problems. ADINA is an unsteady Reynolds averaged Navier Stokes (RANS) solver. For the high Reynolds number case we used Re = A set of finite element models of the system was constructed. he steady-state solution for the nominal operating point of this system (u =, v =.) was obtained using an 88-node mesh constructed in a non-uniform manner so as to increase the model accuracy in the regions of primary interest. Mesh refinements were carried out to obtain a converged solution. In particular, ADINA simulation runs were performed for N = 88 nodes, ~N, ~N, ~4N, ~5N, ~7N, ~8N, ~9N, ~N, ~N, and ~5N. he sample result for the flow field for ~N is shown in Figure 4. he corresponding velocity vectors of the solution with zooming in on the vicinity of the step are shown in Figure 5. It is evident from our results that for the high- fidelity solutions (~N, ~5N) a second recirculation zone has formed at the corner of the Figure. Benchmark flow control problem: D backwardfacing step. his example will illustrate separation control for the D case. he idea is to control the location of the reattachment point on the lower wall using (distributed) measurements of velocity (from micro-electro-mechanical (MEMS) sensors) along the lower wall. he actuator is the suction/blowing, u. he dimensions shown are in meters and the velocity is in units of m/s, the kinematic viscosity, µ, has units of kg/ (m.s), and the density,, has units of kg/m. In this paper we consider both a disturbance rejection as well as a tracking control problem. he control problem is defined as follows. Assume the nominal value of v = m/sec and u ranges between ±.m/sec. We run the Navier- Stokes simulation to steady-state to establish the streamlines and determine the location of the (nominal) reattachment point on the lower wall. For disturbance rejection, we introduce a constant disturbance onto v so that v = + d. We use feedback from the measurement of the reattachment point, y, to the control, u, to move the reattachment point back to its original position. For tracking the reattachment point, we let d =. We command the reattachment point as desired to fore/aft of the nominal reattachment point. We wish to move the reattachment point up or down and hold it at a desired position. We will now address the nonlinear and Figure 4. Flow field for 94 nodes (~N) with reattachment point = 4.58m (for Re = ). Figure 5. Velocity vectors for 94 nodes (~N) with reattachment point = 4.58m (for Re = ). 767

3 step as shown in Figure 5. he velocity profiles were computed for various y distances. By comparing velocity profiles with mesh sizes of ~N and ~5N, we concluded that mesh convergence has been reached with size ~N. o study the properties of the system at low Reynolds numbers, the input velocity v was lowered. he new nominal operating point of this system is (u =, v = -4 m/s) which corresponds to Re =. An 89-node mesh was constructed with higher mesh density in the regions of primary interest in order to increase the model accuracy. o validate the nonlinear model, a series of mesh refinements were carried out again to obtain a converged solution. ADINA simulation runs were performed for N = 89 nodes, ~N, ~4N, ~8N, and ~N. he nature of the solution does not change as the number of the mesh nodes is increased indicating mesh convergence. he velocity profiles were computed for locations y = m, m, m, and 4m measured from the step and are shown in Figure 6 for the case Re =. In the region of interest, where the reattachment point forms (m < y < m and m < z <.5m), the 89-node point mesh solution is virtually identical to the higher order mesh solutions. o ensure mesh independence in the region of interest (the location of the reattachment point), the mesh in that area had higher resolution. In the upper portion of the flow the mesh was quite coarse, especially at N mesh points, which resulted in a solution that was not very mesh independent. he results showed that such errors do not affect the solution in the region of interest. he open-loop disturbance response of the system is shown in Figure Velocity Profile at y = - 4 x -4 Velocity Profile at y = - 4 x Velocity Profile at y = - 4 x -4 Velocity Profile at y = 4-4 x -4 N nodes N nodes 4N nodes 8N nodes N nodes Figure 6. Nonlinear model validation: Velocity profiles at y = m, m, m, and 4m (Re = ). IV. GENERAION OF LINEAR MODELS We define the system state vector as x = [v y t, v z ], where v y is the vector of fluid velocity in the y-direction as measured at each node excepting the nodes embedded in the walls and v z is the z-velocity vector. Notice that the pressure state variables were not included since pressure does not affect the velocity state variables (incompressible fluid). he system Input Velocity v [m/s] Reattach Pt. Loc. [m] Vel. at Reattach Pt [m/s].4 x Open Loop Step Response x ime [sec] Figure 7. Open-loop step response (a) input disturbance signal (top plot), (b) reattachment point location (middle plot), and (c) velocity at the nominal reattachment point (bottom plot). input vector is defined as u = [v, u]. he linear system output is the velocity vector v y, which can in turn be used to calculate the flow reattachment point (through a nonlinear relationship). In order to build a discrete nonlinear model of the system, first the sample period was chosen to be much less than the Courant number of this system [9]. Since the smallest nodal separation was approximately.m in the y-direction and the largest velocity in this direction was approximately.m/sec, dt =.sec was chosen to ensure numerical stability A high-order linear model was obtained by numerical evaluation about the equilibrium (steady-state) solution x, u. By defining the state and input vectors as perturbations from the equilibrium (steady-state) solution, the linear discrete model can be written in standard form: xk Axk Buk yk vy= Cxk [ I ] xk () o validate the linear models, a Pseudo-Random Binary Sequence (PRBS) [5] was selected to excite both the linear and nonlinear models. he outputs of the two models were compared at many nodes. Figure 8 shows a typical comparison plot. From the results we concluded that the high-order linear and the nonlinear models were consistent. Figure 9 shows the pole locations inside the unit circle for the high Reynolds number case. For this case the discrete time sampling period is s =. sec. he locations of the discrete poles range from.9754 to and the damping coefficients range from =.69 to unity. he linear system is non-minimum phase and has a transmission zero (not shown in Figure 9) on the real axis just outside the unit circle. Figure shows the pole locations for Re = case. he locations of the discrete poles range from.9964 to 6.58e-7 and the damping coefficients range from =.747 to unity. he system is minimum phase for this case. 768

4 .6 Linear to Nonlinear comparison of y velocity at Node #98 to PRBS input V Nonlinear Linear reduction, the controllability and observability Gramians (P, Q respectively) were computed [7]-[8]: y y =.5, z = ime [sec] Figure 8. Comparison of the linear and nonlinear model responses at state variable 98 for PRBS excitation in v (for Re = )..5 Poles Inside he Unit Circe APA P BB () A QA Q C C he Hankel singular values of the system were computed and the normalized values are plotted in Figure. σ i, Enr i/ i ( PQ) () Hankel Singular Values, Norm of Reduction Error Hankel sv Norm of Error..5 Imag (z) Order r Figure. Hankel singular values and norm of reduction error (Re = ) Real (z) Figure 9. System poles inside the unit circle (Re = ). Imag (z) Poles Inside he Unit Circle Real (z) Figure. System poles inside the unit circle (Re = ). V. PLAN MODEL ORDER REDUCION wo methods were used for model order reduction: balanced truncation and system identification from simulation data. As a means of evaluating the potential for model-order Assume that an r th order reduced order model was selected from the balanced system, Gr () z C( zi A) B (4) then the reduction error is En r( z) G( z) Gr( z) (5) where Gz ( ) C( zi A) B (6) and for any r, r n, n E ( z) (7) n r i r (and strict inequality holds if i i for any i, r i n ). he reduction error was computed and is also displayed in Figure. he Hankel singular values were also computed for the low Reynolds number case (Re = ), and show a very similar trend. In both cases, amazingly (!) the results suggest a fourth-order reduced plant model. Reduced order plant models were also derived using linear system identification with simulation data. In our case, we excited the high-order linear model with persistently exciting input signal consisting of a combination of steps with a Pseudo-Random-Binary-Sequence (PRBS). We used a prediction error method to identify the dynamics [6]. An autoregressive with an external input (ARX) model structure was chosen and standard routines from the MALAB System Identification toolbox were used. Separate models were identified for the transfer functions i 769

5 from the inputs v and u to the output velocities as recorded along the bottom wall of the modified backward-facing step. All of these transfer functions could be described by either a rd or 4 th order dynamic system. he measurements of the flow velocity directly above the lower wall were used to determine the reattachment point: the point where the velocity is zero. his location was determined by interpolation between the two nodes where the velocity changed sign. We also identified a local transfer function that directly describes the dynamics from v and u to the reattachment point. Again, rd and 4 th order models were sufficient to very accurately predict the local flow behavior. hese system identification results corroborated the results obtained earlier from balanced truncation studies. Hence, we decided to use a 4 th order reduced order model for the feedback regulator designs discussed in the next section. VI. FEEDBACK CONROL SYSEM DESIGN Our goal is to design a robust controller for regulation and tracking. We have used the error space approach for designing robust feedback regulators [5]. A block diagram of the closed-loop system is shown in Figure. shows the open-loop step disturbance response for comparison purposes. he closed-loop system rejects the disturbance rapidly. B. racking Control Reattachment point tracking control was achieved using feedback control for both the high and low Reynolds number cases. he control was used to adjust the input velocity of the u stream to track the desired location. he regulator controllers described above were used to provide feedback control to reduce the velocity at the commanded reattachment point to zero. Figure 5 shows the tracking response of the nonlinear closed-loop system for various reattachment point step commands for the low Reynolds number case (Re = ). It is seen that tracking of the reattachment points is achieved rapidly by the feedback controller using smooth control signals. For this case, the reattachment point was commanded to follow an upward reattachment point disturbance response of the LQG regulator Linear Nonlinear time (sec) Figure. Control structure. Figure. Comparison of the nonlinear and linear regulator closed-loop response for a step disturbance (Re = ). A. Disturbance Rejection A double integral sixth order LQG controller was designed using the error space approach for the case Re = he double integral control was required because of the presence of the non-minimum phase plant transmission zero mentioned before. he controller was tested on the full order (5 state variable) plant model derived in Section IV. A comparison of the response of the linear closed-loop model and the nonlinear ADINA closed-loop model is shown in Figure. It is seen that the linear and nonlinear closed-loop simulations are in good agreement. here is a long tail on the response due to the presence of slow poles. his suggests the potential need for time-optimal control. Velocity at Nominal Reattachment Point [m/sec] 4 x -6 Closed and Open Loop Step Response open-loop closed-loop Closed Loop Response Open Loop Response he design was repeated for the 568 state variable plant model (Re = ) but for a single integral internal model control due to the fact that the system is minimum-phase. A 5 th order internal model controller was designed. he controller was tested on the full order (568 state variable) plant model. Figure 4 shows the disturbance rejection property of the nonlinear closed-loop ADINA model with the disturbance injected at t =, sec. he plot also ime [sec] x 4 Figure 4. Nonlinear regulator closed-loop response with a constant disturbance injected at t =, sec (Re = ), using the LQG controller, and comparison with open-loop disturbance response.

6 step (i.e., move the reattachment point. m to the right), a downward step (i.e., move the reattachment point to the left by.m), and return to the nominal position (i.e., the nominal reattachment point position at y =.57m). As can be seen from Figure 5, the controller quickly follows these commands and moves the reattachment point to its new commanded locations. A MALAB simulation was developed to visualize the closed-loop nonlinear response as shown in Figure 6 where the arrow shows the location of the reattachment point. Reference sig. and reattachment pt. Reference sig. and reattachment pt ime (sec.) x ime (sec.) x 4 Y (input to controller) FB Incremental Control Signal (du) x ime (sec.) x 4 x ime (sec.) x 4 Figure 5. racking response of the nonlinear closed-loop system for reattachment point for various step commands (Re = ). Figure 6. A snapshot of a MALAB movie of the reattachment point tracking response (Re = ). VII. CONCLUSION An effective approach for model reduction for active flow control design was developed. We evaluated various reduction algorithms on high dimensional models for flow over backward-facing step models. he feasibility of the methodology was demonstrated by application of separation control for the D backward-facing step problem. Models were developed for high and low Reynolds numbers. Methods of reducing a high order plant model followed by a low-order controller design were investigated. Plant model order reduction using balanced truncation as well as system identification proved to be very effective yielding a reduction of model order from a full order to a 4 th order reduced order model. Disturbance rejection and tracking of the reattachment point was achieved using LQG controllers. VIII. ACKNOWLEDGEMEN he contributions of Professor Bassam Bamieh of UCSB and Dr. Sarbajit Ghosal of SC Solutions, Inc. are gratefully acknowledged. he authors would also like to thank Dr. James Myatt, Capt. Ian S. Bautista, Mr. Russell Camphouse, and Dr. J. Evers of USAF for their suggestions and feedback. REFERENCES [] M. Gad-el-Hak, Flow Control: Passive, Active, and Reactive Flow Management, Cambridge University Press,. []. R. Bewley, H. Choi, R. emam, and P. Moin, Optimal feedback control of turbulent channel flow, Annual Research Briefs, Center for urbulence Research, Stanford Univ./NASA Ames, 99. [] H. Choi, M. Hinze, and K. Kunisch, Instantaneous Control of Backward-Facing Step Flows, Applied Numerical Mathematics,, pp. -58, 999. [4] P. Moin and. Bewley, Feedback Control of urbulence, Applied Mechanics Reviews, Vol. 47, No. 6, Part, pages S-S; June 994. [5] G. F. Franklin, J. D. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems, 4 th Ed., Prentice-Hall,. [6] B. Bamieh and M. Dahleh, Energy amplification in channel flows with stochastic excitation, Physics of Fluids, (): ,. [7] M. Jovanovic and B. Bamieh, Lyapunov-based distributed control of systems on lattices, submitted to IEEE rans. Automatic Control,. [8] M. Jovanovic and B. Bamieh, Modeling flow statistics using the linearized Navier-Stokes equations, Proc. 4 th IEEE Conference on Decision and Control, Orlando, FL,. [9] M. Jovanovic and B. Bamieh, he spatio-temporal impulse response of the linearized Navier-Stokes equations, Proc. American Control Conference,. [] R. Camphouse and J. Myatt, Feedback Control for a wo- Dimensional Burger s Equation System Model, nd AIAA Flow Control Conference, AIAA-4-4, June 4. [] J. Kim, S. J. Kline, and J. P. Johnston, Investigation of a Reattaching urbulent Shear Layer: Flow over a Backward-facing Step, J. of Fluid Engr., Vol., -8, Vol., Sept. 98. [] J. J. Kim, Investigation of Separation and Reattachment of a urbulent Shear Layer: Flow Over a Backward-Facing Step, Ph.D. Dissertation, Stanford University, April 978. [] H. Le, P. Moin, and J. Kim, Direct Numerical Simulation of urbulent Flow Over a Backward-Facing Step, J. Fluid Mech., vol., pp , 997. [4] V. Michelassi, et al.., Prediction of the backflow and recovery regions in the backward facing step at various Reynolds numbers, Center for urbulence Research, Stanford University, Proc. Summer Program 996. [5] B. Bamieh and M. Dahleh, Energy amplification in channel flows with stochastic excitation, Physics of Fluids, (),. [6] Ljung, System Identification, heory for the User. Prentice-Hall, Englewood Cliffs, NJ, 987. [7] U. M. Al-Saggaf and G. F. Franklin, On Model Reduction, in Proc. 5 th IEEE Conf. Dec. Contr., pp , December 986. [8] U. M. Al-Saggaf, On Model Reduction and Control of Discrete ime Systems, Ph.D. Dissertation, Stanford University, June 986. [9] J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics, Springer,